44 problems

In the orthogonal vector-space situation, let $\binom{x}{n}$ denote the generalized binomial coefficient, let $\boldsymbol{D}_n$ and $\boldsymbol{B}_n$ denote the corresponding Dyn…

Let $A$ be the path algebra of an ad-quiver $(Q,\sigma)$ without oriented cycles. Let $\beta\in\Delta(Q)^+$ be a positive root, and let $p_F(M)$ and $p_\sigma(\beta)$ denote the pe…

Nekrasov's wheel-condition generation conjecture. The localized wheel-condition algebra equals the localized spherical subalgebra:

Let $K$ be a knot with corresponding quiver $Q_K$. For each crossing of type $L_+$, let $K_{i,\tau}$, with $\tau\binom{}{}$ replaced by $\tau eq$? be the knot or link obtained by ch…

Let $Q$ be a finite quiver, let $\widehat{Q}$ denote its Kronecker square, and let $L_{\mathbb K}(Q)$ be the Leavitt path algebra over a field $\mathbb K$. Let…

Let $Q$ be a quiver, let $\widehat{Q}$ denote its Kronecker square, and let $L_{\mathbb K}(Q)$ be the Leavitt path algebra over a field $\mathbb K$. Let…

Let $\Lambda$ be a finite dimensional algebra. Let $\mathsf{W}\subseteq\operatorname{mod}\Lambda$ be a $\tau$-perpendicular subcategory of $\operatorname{mod}\Lambda$, and let…

The ordered-quiver fruitfulness conjecture. The quivers $Q_n$ provide fruitful seeds, and any quiver obtained by gluing them according to an ideal triangulation of a bordered surfa…

The decorated-tiling fruitfulness conjecture. Any purely noncommutative decorated tiling of a polygon provides a fruitful seed. More generally, a decorated tiling of an arbitrary s…

Let $n$ and $p$ be as in the descent algebra setting, let mathop{\Lambda}\nolimits^+_p(n) denote the indexing set of $p$-regular partitions, and let $Q_{n,p}$ be the Ext quiver o…

Shuffle-algebra image conjecture.

Let $Q=(Q_0,Q_1)$ be a quiver, let $\lambda\in\mathbf{C}^{Q_0}$, and let $\Pi_Q^{\lambda}$ be the $\lambda$-deformed preprojective algebra. Write…

Weekly green path conjecture. There is a weekly green path from $v_0$ to $v$. In particular, the products

Let $G=\operatorname{SL}_n(\mathbb{C})$, let $v\leq w$ be elements of the Weyl group $W$, and let $Q_{\operatorname{Lec}}$ and $Q_{\operatorname{Ing}}$ be the quivers constructed f…

Let $Q$ be a quiver, let ${\mathcal B}_{\b0}$ be the slope-zero algebra graded by dimension vectors, and write … where $A_{Q,\boldsymbol{n}}(t)$ is the Kac polynomial. Define … and…

Let $K$ be a knot admitting the standard knots-quivers correspondence, so that the node variables satisfy $x_i\sim x$. Let $Q$ be a quiver corresponding to $K$, with one spectator…

Generalized-quiver partition-function conjecture. After this substitution, the generalized-quiver partition function equals the generating function of Gromov-Witten invariants coun…

Let $Q$ be a quiver, with doubled quiver $Q^d$ and tripled quiver $\widetilde{Q}$. Let the torus $\mathbb{C}^*$ scale the linear maps corresponding to edges of $Q^d$ with weight…

Attractor invariants conjecture. The attractor invariants are given by

Hayashi's quiver quotient conjecture. Every finite-dimensional weak bialgebra $H$ with commutative counital subalgebras is isomorphic to a weak bialgebra quotient of $\hay$ for som…

Isomorphism conjecture. The map $\varphi$ is an isomorphism, equivalently

Hypergeometric contribution conjecture. The ordered multi-degree contributions satisfy

Density conjecture for $X_6$ and $X_7$. The quiver $X_6$ is cluster-$\mathbf{g}$-dense, and the quiver $X_7$ is half cluster-$\mathbf{g}$-dense.

Following Fock and Goncharov, consider a triangulation $\Sigma_0$ and its subtriangulation into sub-triangles, with the associated extended quiver $\mathscr Q$. The nodes of…

Let $A=KQ/{\mathcal I}$ be a finite-dimensional algebra presented by a quiver with relations. Call $A$ minimal if ${\mathcal I}=0$, or if every generating set of relations satisfie…